High-Dimensional Variable Selection for Competing Risks with Cooperative Penalized Regression

Motivation

• Setting: High-dimensional survival data w/ competing risks
  • e.g. time to death from bladder cancer vs. death from other cause
• Typical approach:
  • Fit cause-specific model(s) for event(s) of interest
  • Treats other events as censored
  • → lose (potential) shared information

Building Blocks

1. Penalized Cox regression
   • Elastic net / feature-weighted elastic net (fwelnet)
2. An iterative Algorithm
   • Adapted from multi-task algorithm by fwelnet authors
3. Assumption of shared information between causes
   • Idea: Some features predictive for event 1 will also be predictive for event 2

Foundation: Elastic Net

The elastic net objective function with some negative log-likelihood term:

\underset{\beta}{\operatorname{arg min}} \quad \mathrm{NLL}(\beta) + {\color{blue}\lambda} \sum_{j=1}^p \left( {\color{red}\alpha} |\beta_j| + \frac{1 - {\color{red}\alpha}}{2} \beta^2_j \right)

Feature-Weighted Elastic Net (fwelnet)

• Motivation: Using external information
• Adjust penalization weights on individual or groups of features
• Assign weights / groups via matrix \mathbf{Z} \in \mathbb{R}^{p \times K}

Feature Weighting: Groups

Example for p = 5 features X_{1,2,3,4,5} and K = 2 groups

Feature Weighting: Individually

Example for p = 4 features X_{1,2,3,4}:

Feature-Weighting: New Objective Function

\underset{\beta}{\operatorname{arg min}} \quad \mathrm{NLL}(\beta) + \lambda \sum_{j=1}^p {\color{blue}w_j(\theta)}\left( \alpha |\beta_j| + \frac{1 - \alpha}{2} \beta^2_j \right)

Penalization Weights

• w_j(\theta) is chosen heuristically by the authors for desirable properties
• \mathbf{z}_j^T \theta acts as a score
  • = 0, reduces to original elastic net
  • Higher score \rightarrow lower w_j, feature is “more important”

Feature-Weighting: Single Group

• \mathbf{Z} \in \mathbb{R}^{p \times 1}: No groups, just weights
• Simulation from Tay et al. (2023): \mathbf{Z} set to noisy version of true |\beta|

Transfer to Competing Risks

• Setting with two outcomes/event types: (\mathbf{t}_1, \delta_1), (\mathbf{t}_2, \delta_2)
• Assumption: Shared information for both causes:
  • If X_j is important for cause 1, may also be relevant for cause 2
  • \Rightarrow lower its penalty in cause-specific models
• Basic idea: Adapt previous algorithm to Cox regression
• Multi-task \simeq “Multi-cause”

High-Dimensional Variable Selection

• Simulation setup borrowed from Binder et al. (2009)
• Context: Gene expression data with competing risk target
• n = 400, p = 5000, organized in 4 main blocks
  • 250 variables per block w/ true effects
  • 4 informative variables per block, 16 total
• Fit models, use \mathbf{1}\{\hat{\beta}_j \neq 0\} as classification decision

Simulation of True Effects

• Block 1 (Mutual): \rho \approx 0.5
  same effect (0.5) on both causes

• Block 2 (Reversed): \rho \approx 0.35
  positive effect on cause 1 and negative on cause 2

• Block 3 (Disjoint): \rho \approx 0.05
  3.1: effect on cause 1 only
  3.2: effect on cause 2 only

• Block 4 (Cor. Noise): 500 variables, \rho \approx 0.32
  

• Remaining variables: Uncorrelated noise

Positive Predictive Value

Positive Predictive Value

False Positive Rate

False Positive Rate

How about real data?

• Still WIP
• Idea:
  1. Use algorithms for variable selection
  2. Fit standard cause-specific Cox model using only selected variables
  3. Evaluate prediction performance (tBrier, tAUC)
• Tried bladder cancer data (Dyrskjøt et al. (2005)), did not go well

What’s Next

• Would be nice to have a “working” real data example (Maybe TCGA)
• Finishing the paper for journal submission
• Talk about this at CEN in September